Abstract

In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and nor-malized vectors. It encompasses the already existing notions of unbiasedness for or-thonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measure-ments. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d-dimensional fiducial states are param-eterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum un-certainty with regard to several choices of d+ 1 orthonormal bases are shown, in every dimension d. These last families contain all existing fiducial states in every finite di-mension, and the bases include maximal sets of d + 1 mutually unbiased bases, when d is a prime number.